{"id":3267,"date":"2024-06-18T14:36:19","date_gmt":"2024-06-18T14:36:19","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3267"},"modified":"2024-08-05T02:17:55","modified_gmt":"2024-08-05T02:17:55","slug":"derivatives-and-the-shape-of-a-graph-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/derivatives-and-the-shape-of-a-graph-learn-it-2\/","title":{"raw":"Derivatives and the Shape of a Graph: Learn It 2","rendered":"Derivatives and the Shape of a Graph: Learn It 2"},"content":{"raw":"<h2>Concavity and Points of Inflection<\/h2>\r\n<p id=\"fs-id1165042517819\">We now know how to determine where a function is increasing or decreasing. However, there is another issue to consider regarding the shape of the graph of a function. If the graph curves, does it curve upward or curve downward? This notion is called the <strong>concavity<\/strong> of the function.<\/p>\r\n<p id=\"fs-id1165043256949\">Figure 5(a) shows a function [latex]f[\/latex] with a graph that curves upward. As [latex]x[\/latex] increases, the slope of the tangent line increases. Thus, since the derivative increases as [latex]x[\/latex] increases, [latex]f^{\\prime}[\/latex] is an increasing function. We say this function [latex]f[\/latex] is <strong>concave up<\/strong>.<\/p>\r\n<p>Figure 5(b) shows a function [latex]f[\/latex] that curves downward. As [latex]x[\/latex] increases, the slope of the tangent line decreases. Since the derivative decreases as [latex]x[\/latex] increases, [latex]f^{\\prime}[\/latex] is a decreasing function. We say this function [latex]f[\/latex] is<strong> concave down<\/strong>.<\/p>\r\n\r\n[caption id=\"attachment_3278\" align=\"alignnone\" width=\"633\"]<img class=\"wp-image-3278 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/18172031\/5c0c6292e60d9ab5f29417baa39af97e6ff3c053.png\" alt=\"This figure is broken into four figures labeled a, b, c, and d. Figure a shows a function increasing convexly from (a, f(a)) to (b, f(b)). At two points the derivative is taken and both are increasing, but the one taken further to the right is increasing more. It is noted that f\u2019 is increasing and f is concave up. Figure b shows a function increasing concavely from (a, f(a)) to (b, f(b)). At two points the derivative is taken and both are increasing, but the one taken further to the right is increasing less. It is noted that f\u2019 is decreasing and f is concave down. Figure c shows a function decreasing concavely from (a, f(a)) to (b, f(b)). At two points the derivative is taken and both are decreasing, but the one taken further to the right is decreasing less. It is noted that f\u2019 is increasing and f is concave up. Figure d shows a function decreasing convexly from (a, f(a)) to (b, f(b)). At two points the derivative is taken and both are decreasing, but the one taken further to the right is decreasing more. It is noted that f\u2019 is decreasing and f is concave down.\" width=\"633\" height=\"673\" \/> Figure 5. (a), (c) Since [latex]f^{\\prime}[\/latex] is increasing over the interval [latex](a,b)[\/latex], we say [latex]f[\/latex] is concave up over [latex](a,b)[\/latex]. (b), (d) Since [latex]f^{\\prime}[\/latex] is decreasing over the interval [latex](a,b)[\/latex], we say [latex]f[\/latex] is concave down over [latex](a,b)[\/latex].[\/caption]\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>concave up and concave down<\/h3>\r\n<p>Let [latex]f[\/latex] be a function that is differentiable over an open interval [latex]I[\/latex].<\/p>\r\n<ul>\r\n\t<li>If [latex]f^{\\prime}[\/latex] is increasing over [latex]I[\/latex], we say [latex]f[\/latex] is <strong>concave up<\/strong> over [latex]I[\/latex].<\/li>\r\n\t<li>If [latex]f^{\\prime}[\/latex] is decreasing over [latex]I[\/latex], we say [latex]f[\/latex] is <strong>concave down<\/strong> over [latex]I[\/latex].<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>In general, without having the graph of a function [latex]f[\/latex], how can we determine its concavity?<\/p>\r\n<p>By definition, a function [latex]f[\/latex] is concave up if [latex]f^{\\prime}[\/latex] is increasing. From Corollary 3, we know that if [latex]f^{\\prime}[\/latex] is a differentiable function, then [latex]f^{\\prime}[\/latex] is increasing if its derivative [latex]f^{\\prime \\prime}(x)&gt;0[\/latex]. Therefore, a function [latex]f[\/latex] that is twice differentiable is concave up when [latex]f^{\\prime \\prime}(x)&gt;0[\/latex].<\/p>\r\n<p>Similarly, a function [latex]f[\/latex] is concave down if [latex]f^{\\prime}[\/latex] is decreasing. We know that a differentiable function [latex]f^{\\prime}[\/latex] is decreasing if its derivative [latex]f^{\\prime \\prime}(x)&lt;0[\/latex]. Therefore, a twice-differentiable function [latex]f[\/latex] is concave down when [latex]f^{\\prime \\prime}(x)&lt;0[\/latex].<\/p>\r\n<p>Applying this logic is known as the<strong> concavity test.<\/strong><\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3 style=\"text-align: left;\">test for concavity<\/h3>\r\n<p id=\"fs-id1165043088519\">Let [latex]f[\/latex] be a function that is twice differentiable over an interval [latex]I[\/latex].<\/p>\r\n<ol id=\"fs-id1165042925616\" style=\"list-style-type: lower-roman;\">\r\n\t<li>If [latex]f^{\\prime \\prime}(x)&gt;0[\/latex] for all [latex]x \\in I[\/latex], then [latex]f[\/latex] is concave up over [latex]I[\/latex].<\/li>\r\n\t<li>If [latex]f^{\\prime \\prime}(x)&lt;0[\/latex] for all [latex]x \\in I[\/latex], then [latex]f[\/latex] is concave down over [latex]I[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>\r\n<p id=\"fs-id1165043195362\">We conclude that we can determine the concavity of a function [latex]f[\/latex] by looking at the second derivative of [latex]f[\/latex]. In addition, we observe that a function [latex]f[\/latex] can switch concavity (Figure 6). However, a continuous function can switch concavity only at a point [latex]x[\/latex] if [latex]f^{\\prime \\prime}(x)=0[\/latex] or [latex]f^{\\prime \\prime}(x)[\/latex] is undefined.\u00a0<\/p>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"731\"]<img id=\"16\" src=\"https:\/\/openstax.org\/apps\/archive\/20210421.141058\/resources\/7abb392a249b7e8a60b3285996e777d73855d665\" alt=\"A sinusoidal function is shown that has been shifted into the first quadrant. The function starts decreasing, so f\u2019 &lt; 0 and f\u2019\u2019 &gt; 0. The function reaches the local minimum and starts increasing, so f\u2019 &gt; 0 and f\u2019\u2019 &gt; 0. It is noted that the slope is increasing for these two intervals. The function then reaches an inflection point (a, f(a)) and from here the slop is decreasing even though the function continues to increase, so f\u2019 &gt; 0 and f\u2019\u2019 &lt; 0. The function reaches the maximum and then starts decreasing, so f\u2019 &lt; 0 and f\u2019\u2019 &lt; 0.\" width=\"731\" height=\"401\" \/> Figure 6. Since [latex]f^{\\prime \\prime}(x)&gt;0[\/latex] for [latex]x&lt;a[\/latex], the function [latex]f[\/latex] is concave up over the interval [latex](\u2212\\infty,a)[\/latex]. Since [latex]f^{\\prime \\prime}(x)&lt;0[\/latex] for [latex]x&gt;a[\/latex], the function [latex]f[\/latex] is concave down over the interval [latex](a,\\infty)[\/latex]. The point [latex](a,f(a))[\/latex] is an inflection point of [latex]f[\/latex].[\/caption]\r\n\r\n<p>Consequently, to determine the intervals where a function [latex]f[\/latex] is concave up and concave down, we look for those values of [latex]x[\/latex] where [latex]f^{\\prime \\prime}(x)=0[\/latex] or [latex]f^{\\prime \\prime}(x)[\/latex] is undefined. When we have determined these points, we divide the domain of [latex]f[\/latex] into smaller intervals and determine the sign of [latex]f^{\\prime \\prime}[\/latex] over each of these smaller intervals.<\/p>\r\n<p>If [latex]f^{\\prime \\prime}[\/latex] changes sign as we pass through a point [latex]x[\/latex], then [latex]f[\/latex] changes concavity. It is important to remember that a function [latex]f[\/latex] may not change concavity at a point [latex]x[\/latex] even if [latex]f^{\\prime \\prime}(x)=0[\/latex] or [latex]f^{\\prime \\prime}(x)[\/latex] is undefined. If, however, [latex]f[\/latex] does change concavity at a point [latex]a[\/latex] and [latex]f[\/latex] is continuous at [latex]a[\/latex], we say the point [latex](a,f(a))[\/latex] is an <strong>inflection point<\/strong> of [latex]f[\/latex].<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3 style=\"text-align: left;\">inflection point<\/h3>\r\n<p>If [latex]f[\/latex] is continuous at [latex]a[\/latex] and [latex]f[\/latex] changes concavity at [latex]a[\/latex], the point [latex](a,f(a))[\/latex] is an inflection point of [latex]f[\/latex].<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>For the function [latex]f(x)=x^3-6x^2+9x+30[\/latex], determine all intervals where [latex]f[\/latex] is concave up and all intervals where [latex]f[\/latex] is concave down. List all inflection points for [latex]f[\/latex]. Use a graphing utility to confirm your results.<\/p>\r\n\r\n[reveal-answer q=\"fs-id1165043306699\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165043306699\"]\r\n\r\n<p id=\"fs-id1165043306699\">To determine concavity, we need to find the second derivative [latex]f^{\\prime \\prime}(x)[\/latex].<\/p>\r\n<p>The first derivative is [latex]f^{\\prime}(x)=3x^2-12x+9[\/latex], so the second derivative is [latex]f^{\\prime \\prime}(x)=6x-12[\/latex].<\/p>\r\n<p>If the function changes concavity, it occurs either when [latex]f^{\\prime \\prime}(x)=0[\/latex] or [latex]f^{\\prime \\prime}(x)[\/latex] is undefined. Since [latex]f^{\\prime \\prime}[\/latex] is defined for all real numbers [latex]x[\/latex], we need only find where [latex]f^{\\prime \\prime}(x)=0[\/latex].<\/p>\r\n<p>Solving the equation [latex]6x-12=0[\/latex], we see that [latex]x=2[\/latex] is the only place where [latex]f[\/latex] could change concavity.<\/p>\r\n<p>We now test points over the intervals [latex](\u2212\\infty ,2)[\/latex] and [latex](2,\\infty)[\/latex] to determine the concavity of [latex]f[\/latex]. The points [latex]x=0[\/latex] and [latex]x=3[\/latex] are test points for these intervals.<\/p>\r\n<table id=\"fs-id1165042960171\" class=\"unnumbered\" summary=\"This table has three rows and four columns. The first row is a header row, and it reads from left to right Interval, Test Point, Sign of f\u2019\u2019(x) = 6x \u2013 12 at Test Point, and Conclusion. Below the header, the first column reads (\u2212\u221e, 2) and (2, \u221e). The second column reads x = 0 and x = 3. The third column reads \u2212 and +. The fourth column reads f is concave down and f is concave up.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Interval<\/th>\r\n<th>Test Point<\/th>\r\n<th>Sign of [latex]f^{\\prime \\prime}(x)=6x-12[\/latex] at Test Point<\/th>\r\n<th>Conclusion<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex](\u2212\\infty ,2)[\/latex]<\/td>\r\n<td>[latex]x=0[\/latex]<\/td>\r\n<td>[latex]-[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is concave down<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex](2,\\infty )[\/latex]<\/td>\r\n<td>[latex]x=3[\/latex]<\/td>\r\n<td>[latex]+[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is concave up.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165042329993\">We conclude that [latex]f[\/latex] is concave down over the interval [latex](\u2212\\infty ,2)[\/latex] and concave up over the interval [latex](2,\\infty)[\/latex]. Since [latex]f[\/latex] changes concavity at [latex]x=2[\/latex], the point [latex](2,f(2))=(2,32)[\/latex] is an inflection point. Figure 7\u00a0confirms the analytical results.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"379\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210931\/CNX_Calc_Figure_04_05_011.jpg\" alt=\"The function f(x) = x3 \u2013 6x2 + 9x + 30 is graphed. The inflection point (2, 32) is marked, and it is roughly equidistant from the two local extrema.\" width=\"379\" height=\"385\" \/> Figure 7. The given function has a point of inflection at [latex](2,32)[\/latex] where the graph changes concavity.[\/caption]\r\nWatch the following video to see the worked solution to this example.<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/PBeo4ZJ-FGY?controls=0&amp;start=600&amp;end=713&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end. You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.5DerivativesAndTheShapeOfAGraph600to713_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"4.5 Derivatives and the Shape of a Graph\" here (opens in new window)<\/a>.[\/hidden-answer]<\/section>\r\n<section class=\"textbox proTip\">\r\n<p id=\"fs-id1165043163933\">The table and figure below summarize how the first and second derivatives of a function [latex]f(x)[\/latex] inform the characteristics of its graph.<\/p>\r\n<table id=\"fs-id1165043353366\" summary=\"This table has five rows and four columns. The first row is a header row, and it reads from left to right Sign of f\u2019, Sign of f\u2019\u2019, Is f increasing or decreasing?, and Concavity. Below the header, the first column reads Positive, Positive, Negative, and Negative. The second column reads Positive, Negative, Positive, and Negative. The third column reads Increasing, Increasing, Decreasing, and Decreasing. The fourth column reads Concave up, Concave down, Concave up, and Concave down.\">\r\n<caption>What Derivatives Tell Us about Graphs<\/caption>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Sign of [latex]f^{\\prime}[\/latex]<\/th>\r\n<th>Sign of [latex]f^{\\prime \\prime}[\/latex]<\/th>\r\n<th>Is [latex]f[\/latex] increasing or decreasing?<\/th>\r\n<th>Concavity<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>Positive<\/td>\r\n<td>Positive<\/td>\r\n<td>Increasing<\/td>\r\n<td>Concave up<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Positive<\/td>\r\n<td>Negative<\/td>\r\n<td>Increasing<\/td>\r\n<td>Concave down<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Negative<\/td>\r\n<td>Positive<\/td>\r\n<td>Decreasing<\/td>\r\n<td>Concave up<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Negative<\/td>\r\n<td>Negative<\/td>\r\n<td>Decreasing<\/td>\r\n<td>Concave down<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"937\"]<img id=\"20\" src=\"https:\/\/openstax.org\/apps\/archive\/20210421.141058\/resources\/b4fd75d3ef022de52e9f3b782e4d8908f12a7e23\" alt=\"A function is graphed in the first quadrant. It is broken up into four sections, with the breaks coming at the local minimum, inflection point, and local maximum, respectively. The first section is decreasing and concave up; here, f\u2019 &lt; 0 and f\u2019\u2019 &gt; 0. The second section is increasing and concave up; here, f\u2019 &gt; 0 and f\u2019\u2019 &gt; 0. The third section is increasing and concave down; here, f\u2019 &gt; 0 and f\u2019\u2019 &lt; 0. The fourth section is increasing and concave down; here, f\u2019 &lt; 0 and f\u2019\u2019 &lt; 0.\" width=\"937\" height=\"475\" \/> Figure 8. Consider a twice-differentiable function [latex]f[\/latex] over an open interval [latex]I[\/latex]. If [latex]f^{\\prime}(x)&gt;0[\/latex] for all [latex]x \\in I[\/latex], the function is increasing over [latex]I[\/latex]. If [latex]f^{\\prime}(x)&lt;0[\/latex] for all [latex]x \\in I[\/latex], the function is decreasing over [latex]I[\/latex]. If [latex]f^{\\prime \\prime}(x)&gt;0[\/latex] for all [latex]x \\in I[\/latex], the function is concave up. If [latex]f^{\\prime \\prime}(x)&lt;0[\/latex] for all [latex]x \\in I[\/latex], the function is concave down on [latex]I[\/latex].[\/caption]\r\n<\/section>","rendered":"<h2>Concavity and Points of Inflection<\/h2>\n<p id=\"fs-id1165042517819\">We now know how to determine where a function is increasing or decreasing. However, there is another issue to consider regarding the shape of the graph of a function. If the graph curves, does it curve upward or curve downward? This notion is called the <strong>concavity<\/strong> of the function.<\/p>\n<p id=\"fs-id1165043256949\">Figure 5(a) shows a function [latex]f[\/latex] with a graph that curves upward. As [latex]x[\/latex] increases, the slope of the tangent line increases. Thus, since the derivative increases as [latex]x[\/latex] increases, [latex]f^{\\prime}[\/latex] is an increasing function. We say this function [latex]f[\/latex] is <strong>concave up<\/strong>.<\/p>\n<p>Figure 5(b) shows a function [latex]f[\/latex] that curves downward. As [latex]x[\/latex] increases, the slope of the tangent line decreases. Since the derivative decreases as [latex]x[\/latex] increases, [latex]f^{\\prime}[\/latex] is a decreasing function. We say this function [latex]f[\/latex] is<strong> concave down<\/strong>.<\/p>\n<figure id=\"attachment_3278\" aria-describedby=\"caption-attachment-3278\" style=\"width: 633px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3278 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/18172031\/5c0c6292e60d9ab5f29417baa39af97e6ff3c053.png\" alt=\"This figure is broken into four figures labeled a, b, c, and d. Figure a shows a function increasing convexly from (a, f(a)) to (b, f(b)). At two points the derivative is taken and both are increasing, but the one taken further to the right is increasing more. It is noted that f\u2019 is increasing and f is concave up. Figure b shows a function increasing concavely from (a, f(a)) to (b, f(b)). At two points the derivative is taken and both are increasing, but the one taken further to the right is increasing less. It is noted that f\u2019 is decreasing and f is concave down. Figure c shows a function decreasing concavely from (a, f(a)) to (b, f(b)). At two points the derivative is taken and both are decreasing, but the one taken further to the right is decreasing less. It is noted that f\u2019 is increasing and f is concave up. Figure d shows a function decreasing convexly from (a, f(a)) to (b, f(b)). At two points the derivative is taken and both are decreasing, but the one taken further to the right is decreasing more. It is noted that f\u2019 is decreasing and f is concave down.\" width=\"633\" height=\"673\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/18172031\/5c0c6292e60d9ab5f29417baa39af97e6ff3c053.png 633w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/18172031\/5c0c6292e60d9ab5f29417baa39af97e6ff3c053-282x300.png 282w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/18172031\/5c0c6292e60d9ab5f29417baa39af97e6ff3c053-65x69.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/18172031\/5c0c6292e60d9ab5f29417baa39af97e6ff3c053-225x239.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/18172031\/5c0c6292e60d9ab5f29417baa39af97e6ff3c053-350x372.png 350w\" sizes=\"(max-width: 633px) 100vw, 633px\" \/><figcaption id=\"caption-attachment-3278\" class=\"wp-caption-text\">Figure 5. (a), (c) Since [latex]f^{\\prime}[\/latex] is increasing over the interval [latex](a,b)[\/latex], we say [latex]f[\/latex] is concave up over [latex](a,b)[\/latex]. (b), (d) Since [latex]f^{\\prime}[\/latex] is decreasing over the interval [latex](a,b)[\/latex], we say [latex]f[\/latex] is concave down over [latex](a,b)[\/latex].<\/figcaption><\/figure>\n<section class=\"textbox keyTakeaway\">\n<h3>concave up and concave down<\/h3>\n<p>Let [latex]f[\/latex] be a function that is differentiable over an open interval [latex]I[\/latex].<\/p>\n<ul>\n<li>If [latex]f^{\\prime}[\/latex] is increasing over [latex]I[\/latex], we say [latex]f[\/latex] is <strong>concave up<\/strong> over [latex]I[\/latex].<\/li>\n<li>If [latex]f^{\\prime}[\/latex] is decreasing over [latex]I[\/latex], we say [latex]f[\/latex] is <strong>concave down<\/strong> over [latex]I[\/latex].<\/li>\n<\/ul>\n<\/section>\n<p>In general, without having the graph of a function [latex]f[\/latex], how can we determine its concavity?<\/p>\n<p>By definition, a function [latex]f[\/latex] is concave up if [latex]f^{\\prime}[\/latex] is increasing. From Corollary 3, we know that if [latex]f^{\\prime}[\/latex] is a differentiable function, then [latex]f^{\\prime}[\/latex] is increasing if its derivative [latex]f^{\\prime \\prime}(x)>0[\/latex]. Therefore, a function [latex]f[\/latex] that is twice differentiable is concave up when [latex]f^{\\prime \\prime}(x)>0[\/latex].<\/p>\n<p>Similarly, a function [latex]f[\/latex] is concave down if [latex]f^{\\prime}[\/latex] is decreasing. We know that a differentiable function [latex]f^{\\prime}[\/latex] is decreasing if its derivative [latex]f^{\\prime \\prime}(x)<0[\/latex]. Therefore, a twice-differentiable function [latex]f[\/latex] is concave down when [latex]f^{\\prime \\prime}(x)<0[\/latex].<\/p>\n<p>Applying this logic is known as the<strong> concavity test.<\/strong><\/p>\n<section class=\"textbox keyTakeaway\">\n<h3 style=\"text-align: left;\">test for concavity<\/h3>\n<p id=\"fs-id1165043088519\">Let [latex]f[\/latex] be a function that is twice differentiable over an interval [latex]I[\/latex].<\/p>\n<ol id=\"fs-id1165042925616\" style=\"list-style-type: lower-roman;\">\n<li>If [latex]f^{\\prime \\prime}(x)>0[\/latex] for all [latex]x \\in I[\/latex], then [latex]f[\/latex] is concave up over [latex]I[\/latex].<\/li>\n<li>If [latex]f^{\\prime \\prime}(x)<0[\/latex] for all [latex]x \\in I[\/latex], then [latex]f[\/latex] is concave down over [latex]I[\/latex].<\/li>\n<\/ol>\n<\/section>\n<p id=\"fs-id1165043195362\">We conclude that we can determine the concavity of a function [latex]f[\/latex] by looking at the second derivative of [latex]f[\/latex]. In addition, we observe that a function [latex]f[\/latex] can switch concavity (Figure 6). However, a continuous function can switch concavity only at a point [latex]x[\/latex] if [latex]f^{\\prime \\prime}(x)=0[\/latex] or [latex]f^{\\prime \\prime}(x)[\/latex] is undefined.\u00a0<\/p>\n<figure style=\"width: 731px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" id=\"16\" src=\"https:\/\/openstax.org\/apps\/archive\/20210421.141058\/resources\/7abb392a249b7e8a60b3285996e777d73855d665\" alt=\"A sinusoidal function is shown that has been shifted into the first quadrant. The function starts decreasing, so f\u2019 &lt; 0 and f\u2019\u2019 &gt; 0. The function reaches the local minimum and starts increasing, so f\u2019 &gt; 0 and f\u2019\u2019 &gt; 0. It is noted that the slope is increasing for these two intervals. The function then reaches an inflection point (a, f(a)) and from here the slop is decreasing even though the function continues to increase, so f\u2019 &gt; 0 and f\u2019\u2019 &lt; 0. The function reaches the maximum and then starts decreasing, so f\u2019 &lt; 0 and f\u2019\u2019 &lt; 0.\" width=\"731\" height=\"401\" \/><figcaption class=\"wp-caption-text\">Figure 6. Since [latex]f^{\\prime \\prime}(x)&gt;0[\/latex] for [latex]x&lt;a[\/latex], the function [latex]f[\/latex] is concave up over the interval [latex](\u2212\\infty,a)[\/latex]. Since [latex]f^{\\prime \\prime}(x)&lt;0[\/latex] for [latex]x&gt;a[\/latex], the function [latex]f[\/latex] is concave down over the interval [latex](a,\\infty)[\/latex]. The point [latex](a,f(a))[\/latex] is an inflection point of [latex]f[\/latex].<\/figcaption><\/figure>\n<p>Consequently, to determine the intervals where a function [latex]f[\/latex] is concave up and concave down, we look for those values of [latex]x[\/latex] where [latex]f^{\\prime \\prime}(x)=0[\/latex] or [latex]f^{\\prime \\prime}(x)[\/latex] is undefined. When we have determined these points, we divide the domain of [latex]f[\/latex] into smaller intervals and determine the sign of [latex]f^{\\prime \\prime}[\/latex] over each of these smaller intervals.<\/p>\n<p>If [latex]f^{\\prime \\prime}[\/latex] changes sign as we pass through a point [latex]x[\/latex], then [latex]f[\/latex] changes concavity. It is important to remember that a function [latex]f[\/latex] may not change concavity at a point [latex]x[\/latex] even if [latex]f^{\\prime \\prime}(x)=0[\/latex] or [latex]f^{\\prime \\prime}(x)[\/latex] is undefined. If, however, [latex]f[\/latex] does change concavity at a point [latex]a[\/latex] and [latex]f[\/latex] is continuous at [latex]a[\/latex], we say the point [latex](a,f(a))[\/latex] is an <strong>inflection point<\/strong> of [latex]f[\/latex].<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3 style=\"text-align: left;\">inflection point<\/h3>\n<p>If [latex]f[\/latex] is continuous at [latex]a[\/latex] and [latex]f[\/latex] changes concavity at [latex]a[\/latex], the point [latex](a,f(a))[\/latex] is an inflection point of [latex]f[\/latex].<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>For the function [latex]f(x)=x^3-6x^2+9x+30[\/latex], determine all intervals where [latex]f[\/latex] is concave up and all intervals where [latex]f[\/latex] is concave down. List all inflection points for [latex]f[\/latex]. Use a graphing utility to confirm your results.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165043306699\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165043306699\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043306699\">To determine concavity, we need to find the second derivative [latex]f^{\\prime \\prime}(x)[\/latex].<\/p>\n<p>The first derivative is [latex]f^{\\prime}(x)=3x^2-12x+9[\/latex], so the second derivative is [latex]f^{\\prime \\prime}(x)=6x-12[\/latex].<\/p>\n<p>If the function changes concavity, it occurs either when [latex]f^{\\prime \\prime}(x)=0[\/latex] or [latex]f^{\\prime \\prime}(x)[\/latex] is undefined. Since [latex]f^{\\prime \\prime}[\/latex] is defined for all real numbers [latex]x[\/latex], we need only find where [latex]f^{\\prime \\prime}(x)=0[\/latex].<\/p>\n<p>Solving the equation [latex]6x-12=0[\/latex], we see that [latex]x=2[\/latex] is the only place where [latex]f[\/latex] could change concavity.<\/p>\n<p>We now test points over the intervals [latex](\u2212\\infty ,2)[\/latex] and [latex](2,\\infty)[\/latex] to determine the concavity of [latex]f[\/latex]. The points [latex]x=0[\/latex] and [latex]x=3[\/latex] are test points for these intervals.<\/p>\n<table id=\"fs-id1165042960171\" class=\"unnumbered\" summary=\"This table has three rows and four columns. The first row is a header row, and it reads from left to right Interval, Test Point, Sign of f\u2019\u2019(x) = 6x \u2013 12 at Test Point, and Conclusion. Below the header, the first column reads (\u2212\u221e, 2) and (2, \u221e). The second column reads x = 0 and x = 3. The third column reads \u2212 and +. The fourth column reads f is concave down and f is concave up.\">\n<thead>\n<tr valign=\"top\">\n<th>Interval<\/th>\n<th>Test Point<\/th>\n<th>Sign of [latex]f^{\\prime \\prime}(x)=6x-12[\/latex] at Test Point<\/th>\n<th>Conclusion<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex](\u2212\\infty ,2)[\/latex]<\/td>\n<td>[latex]x=0[\/latex]<\/td>\n<td>[latex]-[\/latex]<\/td>\n<td>[latex]f[\/latex] is concave down<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex](2,\\infty )[\/latex]<\/td>\n<td>[latex]x=3[\/latex]<\/td>\n<td>[latex]+[\/latex]<\/td>\n<td>[latex]f[\/latex] is concave up.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165042329993\">We conclude that [latex]f[\/latex] is concave down over the interval [latex](\u2212\\infty ,2)[\/latex] and concave up over the interval [latex](2,\\infty)[\/latex]. Since [latex]f[\/latex] changes concavity at [latex]x=2[\/latex], the point [latex](2,f(2))=(2,32)[\/latex] is an inflection point. Figure 7\u00a0confirms the analytical results.<\/p>\n<figure style=\"width: 379px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210931\/CNX_Calc_Figure_04_05_011.jpg\" alt=\"The function f(x) = x3 \u2013 6x2 + 9x + 30 is graphed. The inflection point (2, 32) is marked, and it is roughly equidistant from the two local extrema.\" width=\"379\" height=\"385\" \/><figcaption class=\"wp-caption-text\">Figure 7. The given function has a point of inflection at [latex](2,32)[\/latex] where the graph changes concavity.<\/figcaption><\/figure>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/PBeo4ZJ-FGY?controls=0&amp;start=600&amp;end=713&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end. You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.5DerivativesAndTheShapeOfAGraph600to713_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;4.5 Derivatives and the Shape of a Graph&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\">\n<p id=\"fs-id1165043163933\">The table and figure below summarize how the first and second derivatives of a function [latex]f(x)[\/latex] inform the characteristics of its graph.<\/p>\n<table id=\"fs-id1165043353366\" summary=\"This table has five rows and four columns. The first row is a header row, and it reads from left to right Sign of f\u2019, Sign of f\u2019\u2019, Is f increasing or decreasing?, and Concavity. Below the header, the first column reads Positive, Positive, Negative, and Negative. The second column reads Positive, Negative, Positive, and Negative. The third column reads Increasing, Increasing, Decreasing, and Decreasing. The fourth column reads Concave up, Concave down, Concave up, and Concave down.\">\n<caption>What Derivatives Tell Us about Graphs<\/caption>\n<thead>\n<tr valign=\"top\">\n<th>Sign of [latex]f^{\\prime}[\/latex]<\/th>\n<th>Sign of [latex]f^{\\prime \\prime}[\/latex]<\/th>\n<th>Is [latex]f[\/latex] increasing or decreasing?<\/th>\n<th>Concavity<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>Positive<\/td>\n<td>Positive<\/td>\n<td>Increasing<\/td>\n<td>Concave up<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Positive<\/td>\n<td>Negative<\/td>\n<td>Increasing<\/td>\n<td>Concave down<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Negative<\/td>\n<td>Positive<\/td>\n<td>Decreasing<\/td>\n<td>Concave up<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Negative<\/td>\n<td>Negative<\/td>\n<td>Decreasing<\/td>\n<td>Concave down<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<figure style=\"width: 937px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" id=\"20\" src=\"https:\/\/openstax.org\/apps\/archive\/20210421.141058\/resources\/b4fd75d3ef022de52e9f3b782e4d8908f12a7e23\" alt=\"A function is graphed in the first quadrant. It is broken up into four sections, with the breaks coming at the local minimum, inflection point, and local maximum, respectively. The first section is decreasing and concave up; here, f\u2019 &lt; 0 and f\u2019\u2019 &gt; 0. The second section is increasing and concave up; here, f\u2019 &gt; 0 and f\u2019\u2019 &gt; 0. The third section is increasing and concave down; here, f\u2019 &gt; 0 and f\u2019\u2019 &lt; 0. The fourth section is increasing and concave down; here, f\u2019 &lt; 0 and f\u2019\u2019 &lt; 0.\" width=\"937\" height=\"475\" \/><figcaption class=\"wp-caption-text\">Figure 8. Consider a twice-differentiable function [latex]f[\/latex] over an open interval [latex]I[\/latex]. If [latex]f^{\\prime}(x)&gt;0[\/latex] for all [latex]x \\in I[\/latex], the function is increasing over [latex]I[\/latex]. If [latex]f^{\\prime}(x)&lt;0[\/latex] for all [latex]x \\in I[\/latex], the function is decreasing over [latex]I[\/latex]. If [latex]f^{\\prime \\prime}(x)&gt;0[\/latex] for all [latex]x \\in I[\/latex], the function is concave up. If [latex]f^{\\prime \\prime}(x)&lt;0[\/latex] for all [latex]x \\in I[\/latex], the function is concave down on [latex]I[\/latex].<\/figcaption><\/figure>\n<\/section>\n","protected":false},"author":15,"menu_order":26,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":652,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3267"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3267\/revisions"}],"predecessor-version":[{"id":3290,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3267\/revisions\/3290"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/652"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3267\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3267"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3267"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3267"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3267"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}